Clean up spills when they happen. If the spill dries, put a little floor stripper and cleaner designed for use on asphalt floors on a clean rag, and wipe it up.
Use a dust mop or a stick vacuum as needed to pick up animal hair, dust balls and crumbs.
Damp mop the asphalt once a week with a mild detergent and cool water. Go back over the floor with the mop after rinsing it to make sure there is no detergent buildup.
Shine the floor with a mixture of 1 cup of fabric softener in half a bucket of water if it becomes dull, or use a commercial product designed to shine asphalt floors.
Clean heavily soiled floors with a floor stripper and cleaner. Follow the manufacturer's instructions.
Remove crayon marks with a clean cloth dampened with mineral spirits. Lipstick marks can be cleaned with a cloth dampened with isopropyl alcohol. Most other stains can be treated with floor stripper and cleaner.